Math CentralQuandaries & Queries


Question from Debbie:

I would like to get a list of all the possible 5 digit combinations of the numbers 12345.
Thank you for any help.

Hi Debbie,

I can show you how to write them down yourself.

First of all how many are there? If you are going to write one of them you have 5 choices as to which digit you write down first. Once you have done that there are 4 digits left so you have 4 choices for the second digit. Now that you have chosen the first 2 digits there are 3 remaining and thus you have 3 choices for the third digit. Hence there are 2 choices for the fourth digit and only 1 choice for the fifth digit. Thus you have made 5 × 4 × 3 × 2 1 = 120 choices and there are 120 possible 5 digit numbers made from 1, 2, 3, 4 and 5 if you don't allow any digit to be repeated.

You can write them down in numerical order from smallest to largest. The smallest is 12345 followed by 12354. These are the only possibilities with 123 in the first 3 places. The next largest will have 124 in the first 3 places. Again in numerical order they are 12435 and 12453. The next largest will have 125 in the first 3 places and they are 12534 and 12543. So far I have listed all of the possibilities with 12 in the first two places. the are


Now consider the possibilities with 13 as the first two digits. Using an argument as above I find 6 more.


In a similar manner there are 6 more with 14 as the first 2 digits and then an additional 6 with 15 as the first 2 digits. This in total gives 6 × 4 = 24 possibilities with 1 as the first digit.

Now start again with 2 as the first digit. The smallest is 21345, the largest is 25431 and again there are 24 possibilities. Repeat this process again with 3 as the first digit, then 4 as the first digit and finally 5 as the first digit. In total you will find 5 × 24 = 120 possibilities.

I hope this helps,

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